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Probability Basics
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Probability
• A numerical measure of the likelihood of occurrence of an event
• Provides description of uncertainty
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Experiments
• A process that generates well-defined outcomes• At each repetition of experiment, one and only
one outcome will occur
Experiment Experimental outcomeFlip a coin Head, tail (side 1, side
2)Play a game Win, lose, tieRoll a dice 1, 2, 3, 4, 5, 6Inspect a part Pass, fail
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Sample Space
• Sample space is the set of all experimental outcomes– Flip a coin: S = {Head, Tail}– Roll a die: S = {1, 2, 3, 4, 5, 6}
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Assigning Probabilities
• Must adhere to two conditions:– The probability of an outcome:
0 ≤ P(Ei) ≥ 1 for all iEi = i th Experimental outcome
– The sum of the probabilities of all outcomes:∑ P(Ei) = 1
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Methods of assigning Probabilities
1. Classical Method– When it is reasonable to assume that all
outcomes are equally likely:P(Ei)= 1/n, n= number of outcomes
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Methods of assigning Probabilities
2. Relative Frequency Method– Relative frequency based on limited
repetition of the experimentIn a sample of 200 parts 10 are defectiveP (defective) = 10/200 = 0.05
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Methods of assigning Probabilities
3. Subjective Method– Used when the classical or the relative
frequency method are not possible, or as a support for these methods
– Probability is determined as the degree of belief that an outcome will occur based on available information and past experience
– Assignment must ensure the two conditions of assignment are applicable
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Events and Their Probabilities
• An event is a collection of outcomes (sample points)– Roll a die experiment: define event A as
outcome is an even number• P (event) = sum of probabilities of sample
pointsP (A) = P(2) + P (4) + P (6)
= 1/6 + 1/6 + 1/6 = ½
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Complement of an Event
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Union of Two Events
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Intersection of Two Events
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Addition Law
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Mutually Exclusive Events
• Events that are mutually exclusive do not share any sample points in common (probability of the intersection is zero)
• Therefore:P (A U B) = P (A) + P (B) – P (A ∩ B)
Becomes P (A U B) = P (A) + P (B)
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Conditional Probability
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Independent Events
• Two events are independent ifP (B/A) = P (B) orP (A/B) = P (A)
[The probability of one does not affect the probability of the other]
• Are mutually exclusive events independent events?
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Example
Suppose we have the following data for why students consider joining a college (Question 13)
Reason for Joining
quality Cost Other Totals
Full time 421 393 76 890
Part time 400 593 46 1039
Totals 821 896 122 1929
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Events
Event student is full time F
Event student is part time T
Event reason is Quality Q
Event reason is Cost C
Event reason is Other O
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Joint Probability Table
Reason for JoiningQuality
(Q)Cost
(C)Other
(O)Totals
Full time(F)
0.22 0.20 0.04 0.46
Part time(T)
0.21 0.31 0.02 0.54
Totals 0.43 0.46 0.06 1.00
Status
Joint probabilities Marginal probabilities
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Probabilities of Events
P (F) =
P (T) =
P (Q) =
P (C) =
P (O) =
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Probabilities of Events
Student is FT ? P (Reason is Quality)Student is PT ? P (Reason is QualityAre T and Q independent?
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Multiplication Law
• Multiplication law is use to find the intersection between two events
P (A ∩ B) = P (A/B) P(B) P (A ∩ B) = P (B/A) P(A)
• If A and B are independent then
P (A ∩ B) = P (A) P(B)
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Bayes’ Theorem
• Bayes’ theorem is used to update the probabilities of events based on new information (a sample, a test, etc.)
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Example
• A factory receives parts from two suppliers– A1 = Event part is from supplier 1 P (A1) = 0.65– A2 = Event part is from supplier 2 P (A2) = 0.35– G = Event part is good– B = Event part is bad
P (G/A1) = 0.98 P (B/A1) = 0.0.02P (G/A2) = 0.98 P (B/A2) = 0.0.02
• Suppose we select a bad part: What is the probability that it came from supplier 1? P (A1/B)What is the probability that it came from supplier 2? P (A2/B)
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Probability Tree for Two-supplier Example
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Bayes’ Theorem: Tabular Procedure
(1) (2) (3) (4) (5)
EventPrior Prob
CondProb
Joint Prob
Posterior Probability
Ai P(Ai) P(B/Ai) P(Ai ∩ B) P(Ai /B)
A1 0.65 0.02 0.0130 0.0130/0.0305 = 0.4262
A2 0.35 0.05 0.0175 0.0175/0.0305 = 0.57381.00 P(B) = 0.0305 1.0000